Mech Fingers mesh folds into itself
If you reparent With Automatic Weight it will works rather fine, so the solution may be to make hands separate objects, parent With Automatic Weight, then if necessary join back to the rest:

If you don't want to reparent With Automatic Weight for some reasons, then you need to give some manual corrections in order to extend a bit the influence of each bone. First thing, display the layers with the bones that deform your mesh and the ones that command:

You need then to extend a bit the influence of each phalanx, before:

After:

It will work better:

Use the Paint Mask option to make it easier to paint on the right faces, also make sure that you enable all the options that allow to paint through (in the N panel):

You can also do all the corrections through the Vertex Groups list and Assign button, which may be quicker: